#7766594

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#47453	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,5,8)(3,6,4,7)(9,10)(11,13)(12,16)(14,18)(15,17)(19,21)(20,23)(22,24)(25,26,31,28)(27,35,36,32)(29,33,40,38)(30,34,37,39);(1,2)(3,6)(4,7)(5,8)(9,11,15,12)(10,16,17,13)(14,19,22,20)(18,23,24,21)(25,27,31,36)(26,32,28,35)(29,37,40,30)(33,34,38,39);(10,17)(11,12)(14,22)(21,23)(27,36)(30,37)(32,35)(34,39);(1,3,5,4)(2,6,8,7)(9,12,15,11)(10,16,17,13)(14,20,22,19)(18,23,24,21)(25,28,31,26)(27,35,36,32)(29,33,40,38)(30,39,37,34);(9,13,15,16)(10,12,17,11)(14,21,22,23)(18,20,24,19)(25,29)(26,33)(27,37)(28,38)(30,36)(31,40)(32,34)(35,39);(1,4,5,3)(2,7,8,6)(9,14,15,22)(10,18,17,24)(11,19,12,20)(13,21,16,23)(25,30,31,37)(26,34,28,39)(27,29,36,40)(32,38,35,33)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7673790);